How to Solve Related Rates Problems: Calculating Egg's Shadow Speed"

[BrownianMan]

One night, a toddler is dropping eggs from a windowof a building which is 20 feet above ground. A neighbor, hearing the commotion, looks through her window which is 15 feet from the toddler's window, on the same wall of the building, and at the same height. She shines a flashlight from her window on one of the falling eggs. How fast is the shadow of this egg moving on the ground when the egg is halfway to the ground?

The prof did this in class and got an answer of 42 m/sec. The answer in the back of the book says 76 ft/sec. I get the same answer as the prof, but I'm not sure if it's correct.

Is the book wrong??

 

[Dick]

I'm agreeing with the book.

 

[BrownianMan]

How did you solve it?

 

[Dick]

How did you solve it?

I drew a diagram, wrote down relations between sides, differentiated etc. This is a HW question. Show your solution and I'll look at it.

 

[BrownianMan]

This is what I did:

I let h(t) be the height of the egg t seconds after being dropped, and let s(t) be distance at t seconds between shadow of egg and the point the egg hits the ground.

So tan(theta) = 20/(15 + s) = h/s. Hence, 20s = 15h + sh, and 20s' = 15h' + sh' + s'h.

h(t) = -4.9t^2 + 20
h'(t) = -9.8t

at h = 10:

10 = -4.9t^2 + 20
t = 10/7

and substituting into 20s = 15h + sh, I get s = 15. Then,

20s' = -15(9.8)(10/7) + 10s' + (-9.8)(10/7)(15)
s' = -42 ft/sec

 

[Dick]

This is what I did:

I let h(t) be the height of the egg t seconds after being dropped, and let s(t) be distance at t seconds between shadow of egg and the point the egg hits the ground.

So tan(theta) = 20/(15 + s) = h/s. Hence, 20s = 15h + sh, and 20s' = 15h' + sh' + s'h.

h(t) = -4.9t^2 + 20
h'(t) = -9.8t

at h = 10:

10 = -4.9t^2 + 20
t = 10/7

and substituting into 20s = 15h + sh, I get s = 15. Then,

20s' = -15(9.8)(10/7) + 10s' + (-9.8)(10/7)(15)
s' = -42 ft/sec

That's pretty much what I did. For one thing the units of the problem are feet. You are using 9.8m/s^2 for g. That's the metric unit. You want 32 ft/s^2 in english units.

 

[BrownianMan]

Ah, silly mistake!

Thanks, it all works now.